It is well known that an electromagnetic signal generally can be expressed as two components—amplitude and phase. Highly complex and sophisticated processing techniques have been developed to extract information from the amplitude and phase of such a signal.
Synthetic Aperture Radar (SAR) is one such technique. It is an active remote sensing technique that utilizes radar satellite images and sophisticated post-processing. Unlike passive sensing, SAR involves the transmission of electromagnetic radiation as a wave from a source toward a target, which, in turn, reflects radiation back toward the source. The reflected radiation returns as a signal that can analyzed in terms of its amplitude components and its phase components. For many SAR systems, only the amplitude data of the return signal is used to extract information; however, increasingly, the phase components of the return signal are being used to enhance the information extracted from the return signal.
Interferometry is a technique that compares the phase and amplitude of two waves. Interferometric SAR (InSAR) is a technique that uses two images of the same area, such as a geographic region, and generates an interferogram from the difference in phase between the two images. The phase difference is measured and is recorded as repeating fringes. For many quantitative applications, the fringes present in the interferogram must be “unwrapped”. Typically, this involves counting the integer number of fringes, which correspond to an integer number of wavelengths, to produce a field of displacement along the “line of sight” between the radar antenna and the ground pixel. The unwrapped interferogram may then be further analyzed, interpreted, and modeled to characterize geophysical changes of the geographic region.
More particularly, InSAR is a geodetic technique that calculates the interference pattern caused by the difference in phase between two images acquired by a space-borne or airborne SAR at two distinct times. The resulting interferogram is a contour map of the change in distance between the ground and the radar instrument. These maps provide an unsurpassed spatial sampling density (of the order of 100 pixels per square kilometer), a competitive precision (of the order of 10 millimeters) and a useful observation cadence (of the order of 1 pass per month). This remote-sensing tool has been demonstrated and validated for many actively deforming areas, including natural earthquakes and anthropogenic activity.
Each time a radar sensor acquires an image, it records, for each pixel, a complex number composed of an amplitude and a phase. The amplitude measures the backscatter or the intensity of electromagnetic radiation reflected by the ground back to the radar antenna. The phase provides a measurement of the round trip travel time (and thus the distance) between the satellite and the ground. Given two such complex images acquired at two distinct times, InSAR provides four data products: two amplitude images, a map of the phase shift (called an interferogram), and a map of the spatial coherence (the degree of resemblance in the phase shift between neighboring pixels). All four of these maps are typically registered to a digital elevation model (DEM) in geographic or cartographic coordinates within a distance of the order of 10 meters.
The InSAR phase values range from −½ to +½ cycles. The phase values are generally ambiguous because they are determined only to within an unknown integer number of cycles. Accordingly, the phase values are typically referred to as “wrapped” phase values. To interpret the wrapped values in terms of range change in millimeters, however, conventional InSAR resolves the integer ambiguities in a process known as “unwrapping”. Unwrapping algorithms have been developed, but these algorithms can break down in areas where the phase is noisy as a result of decorrelation, leading to significant errors in the unwrapped phase data. Few, if any, of these unwrapping algorithms provide an estimate of the uncertainty of their output, preventing any attempt to weight the data in a statistical analysis.
A Global Navigation Satellite System (GNSS) receiver is an example of a device that can use the amplitude and phase components of an electromagnetic signal to determine a position relative to a global coordinate system. Generally, the GNSS is composed of a constellation of orbital satellites capable of communicating with GNSS receivers. The Global Positioning System (GPS) operated by the U.S. government is a subset of the GNSS. A GNSS receiver typically includes software and a user interface designed to determine the position of the GNSS receiver from signals received by the orbital satellites. Generally, each satellite generates signals at one or more frequencies. Each signal is detected and stored by the GNSS receiver for processing, such as by the GNSS receiver or other processors, such as a computer workstation, communicatively linked to the GNSS receiver. More particularly, the GNSS receiver stores data consisting of recordings of the signals transmitted by the satellites.
The GNSS receiver will receive signals from at least four separate orbiting satellites to estimate at least four unknown parameters that define the position of the GNSS receiver. Those parameters include latitude, longitude, elevation, and time. Each satellite can provide several types of signals, including an absolute but imprecise distance called a pseudorange as well as an ambiguous (wrapped) but precise phase measurement. Either observable quantity can be converted to a distance measurement that can be used to estimate the position of the GNSS receiver in a process known as trilateration.
In current practice, to use the more precise phase measurements, the GNSS positioning device must first unwrap the wrapped phase. In the context of GNSS processing, this is commonly referred to as “bias fixing”, “resolving integer ambiguities”, or “fixing cycle-slips”. As in the InSAR example described above, the unwrapping procedure requires significant computing power. Consequently, most consumer-grade GNSS receivers, such as hand-held GPS receivers and vehicle-mounted GPS receivers, use pseudorange measurements, rather than phase measurements, as data to estimate values for the unknown parameters. In this regard, typical consumer-grade GNSS receivers using pseudorange data alone will provide a positional precision of 1 to 10 meters, whereas phase measurements, when properly unwrapped, can yield a positional precision better than 1 meter.